ETD Question

[Richard Brekne]

Marc Damashek wrote:

In answer to your first question below, it is known, and becoming more
widespread knowledge as time goes by. It has be termed para-inharmonicity
by some tho I prefer to use the terms positive and negative inharmonicity
when refering to measureable inharmonicity, and expected inharmonicity when
refering to theoretical inharmonicity.

> 
>  
>         If I were to plot up the number of cents each partial was
> sharp, the points would not lie on a smoothly rising curve, but would
> instead deviate randomly from that curve, with some of the deviations
> being pretty sizable. The Strobotuner showed me that the successively
> faster advance rates of the upper partials on some strings wasn't
> uniform: some partials were obviously slower or faster than they
> ought to have been, and didn't interpolate the frequencies of their
> neighbors.
> 
>         Is this higher-order effect well known? I can't recall ever
> seeing it discussed. It most certainly exists and is one reason why
> every piano has its very own personality. It results from string
> irregularities of one kind or another (density, diameter, etc.,
> including nicks in the string, rust, and so on). Such imperfections
> affect the frequency of different partials differently, depending on
> the detailed physical distribution of the defects. For instance,
> altering a plain wire string right at its midpoint (for instance, by
> nicking it or loading it with some fine wire wrapped tightly around
> it) will not affect the (stretched) even harmonics, which have a node
> at that point. But it will change the frequency of the odd
> (stretched) harmonics, as well as inducing other craziness such as
> false beats between the various vibrating string segments.
> 
>         So the hierarchy of tuning subtlety goes something like this:
> 
>         perfect strings (harmonics at integer multiples of the fundamental)


> 
>         --> strings with inharmonicity (harmonics at successively
> higher frequencies than integer multiples -- stretched scales)
> 
Expected inharmonicity

>         --> strings with inevitable random imperfections (random
> deviations from smoothly stretched scales)

Unexpected, or positive and negative inharmonicity.
> 
>         Every piano tuning is a compromise in which we try to
> accommodate those random deviations as best we can, and that's what
> really separates the sheep from the goats.

True enough, probably more so then even the most informed ears are willing
to give credit for.

> 
>         Now, how do existing ETD's decide what the 'best' frequency
> is for a string? Obviously not by simply targeting the equal-tempered
> fundamentals, because of inharmonicity (first-order correction). But
> do any of them explicitly optimize their targets based on accurate
> measurements of the partials on all strings (at least in octaves 2-5,
> say), which can exhibit random deviations from a smoothly stretched
> scale (second-order correction)? Probably not -- note that this is
> different from measuring just the A's up the keyboard, for instance,
> in order to find an appropriate stretching curve (which is a Good
> Thing as far as it goes).

No.. no such device exists yet, tho you can use both the strobe and tunelab
to do basically this as a tuning procedure. I call it direct partial
referencing, Jim Coleman had another name for this. Unfortunanatly you
cannot practically calculate any kind of optimum curve in this manner. You
can just decide on your own set of priorities for a curve and apply them.
Tunelab however gives you the ability to plot what you have done and adjust
it graphically.
> 
>         While it might seem awfully complicated to obtain a
> self-consistent tuning solution that minimizes the bad effects of
> random irregularities, it's not impossible, especially when you're
> toting around a fast laptop computer and can already accurately
> measure the partials. (The solution involves some straightforward
> linear algebra.) I strongly suspect that it is precisely because
> ETD's ignore random irregularities in real strings that the best
> aural tunings still outstrip the best ETD tunings, especially on the
> worst pianos. If this situation were to change -- and I claim that it
> can -- we would hear the improvement immediately.

I aggree, except that I would point to the best pianos instead of the worst
pianos relating to the comparison you make about aural vs ETD tunings.
Actually I keep thinking that it should be possible to find some way of
combining ETD calculated curves with direct partial referencing to create a
kind of "on the fly" self adjusting curve calculation that would not be so
time consuming as to defeat its pupose.
> 
>         It would be very valuable to get more information from the
> people who have designed and done the programming on the ETD's in
> order to shed some light here. It would also be a whole lot of fun to
> produce a working prototype and set it to work on a piano-shaped
> object.

Talk to Robert Scott, Dean Rayburn and Dr Sanderson... grin...
> 
> Marc Damashek
> Hampstead, MD

-- 
Richard Brekne
Associate PTG, N.P.T.F.
Bergen, Norway